This invention relates to apparatus and methods for controlling the response of a wave energy converter (WEC) so as to increase the efficiency of the WEC in producing energy. The invention also relates to control systems for wave energy conversion devices, and more particularly to a control algorithm for capturing wave energy more efficiently.

Wave energy converters (WECs) of interest generally include first and second bodies which can move relative to each other when placed in a body of water and subjected to the forces of the waves. Typically, one body, referred to as a float or shell, is designed to move in phase with the waves and the other body, referred to as a column or spar, is designed to either move out of phase with the waves (resulting in a two body system) or to be held relatively fixed (resulting in what may be termed as a one body system). A power conversion mechanism (or power take-off, PTO, device) is connected between the two bodies to convert their relative motion into a form of useful energy (e.g., electrical). Typically, WECs of interest include a controller and data processor which may be a part of, or separate from, the PTO for adjusting the operation of the power take-off (PTO) device.

A PTO for use in practicing the invention may include any suitable device or component (e.g., a motor/generator, a linear electric generator, a rack and pinion, or any other mechanisms or linkages) connected between the first and second bodies of a WEC capable of converting their relative motion into useful energy (e.g., electric power). The performance and functioning of the power take-off (PTO) is critical to harnessing wave power in an efficient manner. Generally, the dynamic behavior of a WEC's PTO is governed by a combination of environmental factors and control factors. The environmental factors include wave- and wind-induced forces acting on the bodies of the WEC. The control factors are governed by the WEC's controller and data processor which may be used to adjust force(s) applied to, and between, the first and second bodies of the WEC.

A WEC system can be modeled to generate a mathematical formulation which can be used to predict the responses of the WEC and the PTO. Different WEC and PTO systems may respond in different ways. The way a particular WEC and its PTO respond to various stimuli can be predicted. Thus, the response of a PTO to the relative movement of the two bodies (e.g., float and spar) can be defined or modeled. Likewise, the response (movement) of the WEC to energy supplied to the PTO can also be defined or modeled. The better the model, the more accurate is the mathematically predicted response of the WEC and PTO. The mathematical prediction(s) may be used to control the tuning of the PTO and/or the positioning of the PTO and/or the infusion and extraction of power to and from the PTO to increase the efficiency of the system. This presumes the incorporation of a controller and data processor coupled to, and between, the components (e.g., float, spar, PTO) of the WEC. The behavior of the PTO connected between two bodies which move relative to each other in response to motion of the waves and which converts their relative motion into useful energy is significant in harnessing wave power in an efficient manner.

Typically, power is extracted from the PTO. But, at times, it is desirable to supply power to the PTO and cause the PTO to control or adjust the relative motion of the two bodies to optimize the average power obtained from the relative motion of the two bodies (float and spar). As noted above, the two bodies may be designed to move, or only one may be designed to move with the other body being held fixed.

To optimize the power generated by a WEC, the WEC system may include a wave sensor located "upstream", at some distance from the WEC, to sense selected characteristics of an incoming wave and provide the WEC with "predictive" information as to the incoming wave. The predictive information is desirable since, due to properties of moving floating bodies, optimal power capture requires motion of the WEC and PTO with regard to waves arriving some seconds later. Thus, systems of interest include a wave-sensing (generally external to the WEC) and prediction subsystem for sensing selected characteristics such as the amplitude (elevation) and frequency of incoming waves in order to determine the nature of the waves predicted to impinge on the WEC, and a prediction subsystem. The wave prediction subsystem supplies its signals to a control computer which then controls or adjusts the PTO (which may include supplying and/or extracting power from the WEC) in an effort to try to optimize the average power output of the WEC.

However, a problem exists in that the predicted wave Ψ_P may differ from the actual wave Ψ_A which impinges on the WEC. The difference may be due to several different factors. One factor is that the wave-detection sub-subsystem of the wave prediction subsystem is prone to inaccuracies in sampling, which leads to prediction errors. Also, there are many different currents and atmospheric conditions which cause the actual wave incident on the WEC to be different form the predicted wave. The difference between a predicted wave Ψ_P and the actual wave Ψ_A may be defined as "wave error." If, and when, an actual wave Ψ_A, different from the predicted wave Ψ_P, impinges on the WEC, the WEC components (e.g., the PTO and floating bodies) may not be positioned to their optimum position for maximizing response to the impinging wave (and in fact may be positioned significantly away therefrom). The result is that the response of the WEC is less efficient.

In a known prior art method full dependence is placed on the information corresponding to the predicted wave and the assumption that the predicted wave will impinge on the WEC. This gives rise to significant errors when the wave which actually impinges on the WEC is different than the predicted wave. Therefore, the prior art scheme is not satisfactory. Furthermore, there is no known economically viable system which can provide sufficiently accurate wave prediction. Thus, a problem with known predictive systems occurs when there is a wave error (e.g. when the wave impinging on the WEC is different than the predicted wave.)

Document WO2007/137426A discloses a wave energy converter system according to the preamble of claim 1.

Applicant's invention resides, in part, in the recognition that relying solely on the predicted information associated with an incoming wave does not provide optimal results because the "actual" wave impinging on the wave energy converter (WEC), which determines the actual forces seen by the WEC, may be significantly different than the anticipated "predicted" wave. Thus, relying solely on the predicted information to position and move the WEC components does not compensate for "wave errors" and leads to less than optimal results since the WEC components may not be positioned at their optimum point(s) and do not undergo optimum motions.

Applicant's invention includes a method and apparatus for using a combination of "predictive" and "actual" signals to control the WEC. The combination of signals includes reliance, in part, on the predicted information obtained from a wave sensor and, in part, on the actual response (e.g. relative velocity) of the components of the WEC. An appropriate selection and combination of these signals results in an improved control system with less error and a WEC system yielding greater power efficiency.

A WEC system embodying the invention includes a sensor, generally external to the WEC, for sensing the amplitude (and/or other selected characteristics, such as frequency) of an incoming wave and a wave sensing and responsive.subsystem for supplying corresponding "predictive" signals to a control computer which includes data processing capability. Signals representative of the selected actual conditions (e.g. the velocity) of the WEC are also supplied to the control computer which is programmed to process the predicted and actual information in order to generate appropriate signals (forces) to the components of the WEC such that the average wave power captured by the PTO, and made available to a load, is maximized.

The programming of the control computer may be based on mathematical models of the WEC and its responses to various conditions, as discussed below. In accordance with the invention, based on a mathematical modeling of the WEC, the force F₁ or F_PTO which is the force applied to the PTO by the WEC (when power is being generated by the WEC) and/or the force applied by the PTO to the WEC (when power is being consumed by the WEC), is determined as follows: $${\mathrm{F}}_{\mathrm{PTO}}={\mathrm{F}}_{\mathrm{PTO}}\left(\mathrm{P}\right)-\mathrm{\beta }\left({\mathrm{V}}_{\mathrm{P}}-{\mathrm{V}}_{\mathrm{ACT}}\right);$$ Where:

1. (a) F_PTO(P) is the force applied to the PTO by the WEC and/or the force applied by the PTO to the WEC, based on the predicted wave;
2. (b) V_P is the predicted velocity of the WEC in response to the predicted wave;
3. (c) V_ACT is the "actual" or instantaneous velocity of the WEC which can be determined by sensing a selected point (or points) reflecting the actual (instantaneous) relative motion of the bodies (or body) constituting the WEC; and
4. (d) β is determined by calculating:
   1. i- Z_PTO, which is the selected complex mechanical impedance of the PTO; and
   2. ii- The value of β which will minimize the term (Z _PTO - β) (V_P) for selected conditions of Z_PTO (e.g., the value of Z_PTO for dominant wave frequency), subject to the constraint that β is either constant with respect to ω, or is a causal function with respect to ω.

By way of example, a method embodying the invention includes:

1. (i) Obtaining a mathematical function termed Γ_β(ω) which is a characteristic of the WEC geometry and the dominant wave conditions;
2. (ii) Obtaining wave prediction signals from a wave prediction subsystem;
3. (iii) Obtaining an instantaneous measurement of the actual PTO/WEC velocity and/or position;
4. (iv) Calculating a portion of the desired PTO force (the causal force, F_c) as being proportional to a constant (β) multiplied by the actual PTO velocity;
   where the value of β is selected such that it will minimize the term (Z_PTO - β) for selected conditions of Z _PTO (e.g., the value of Z _PTO for dominant wave frequency), and where Z _PTO may have the characteristics shown in Fig. 9 which shows the real and imaginary parts of the complex PTO impedance for a particular example of a WEC.
5. (v) Calculating a second portion of the desired PTO force (the acausal force, F_A) as being the result of a well-defined mathematical operation depending on the Γ_β(ω) function from step (i), and the wave prediction from step (ii) above;
6. (vi) Summing the two PTO forces to obtain the total desired force (F₁ = F_PTO= F_A+ F_C) ; and
7. (vii) Applying that force to the PTO.

In the accompanying drawings like reference characters denote like components, and

• Figure 1 is a highly simplified block diagram of a WEC suitable for practicing the invention;
• Figure 2 is a flow chart of an "outer loop" control system in accordance with the invention;
• Figure 3 is a flow chart of a "wave prediction" system for use in practicing the present invention
• Figure 4 is a flow chart of an "inner-loop" control system for calculating Fpto in accordance with the invention;
• Figure 5 is a flow diagram showing the processing of information and controls in accordance with one aspect of the invention.
• Figures 6 and 7 are illustrative diagrams for obtaining the actual velocity of the WEC.
• Figure 8 is a waveshape representation of a time-domain optimal force response transfer function according to the invention and according to the prior art;
• Figure 9 shows the functional form of the real and imaginary parts of the optimal PTO impedance (Z_PTO) for a WEC geometry of interest;
• Figure 10 is a frequency domain graph of the functions Γ₀(ω) and Γ_β(ω) for a given value of β;
• Figures 11, 12, 13, and 14 are illustrative waveform diagrams intended to illustrate the benefits of applying the inventive concept to a WEC system
• Figure 15 is a graph displaying results of a numerical model, showing the improved characteristics using the method of the invention relative to a prior art method, with regard to wave error;

Figure 1 shows the main components of a wave energy converter (WEC) system for use in practicing the invention. The WEC includes a first body or float, 100, and a second body or spar, 200. The first body (e.g., float) is designed to move generally in phase with the waves and the second body (e.g., spar) is designed to move generally out of phase with the waves, or to be held relatively fixed at mechanical ground. A power take-off (PTO) device, 300, is coupled between the first and second bodies and converts their relative motion into useful energy and is capable of receiving power from, or transmitting power to, the first and second bodies. The PTO may be any suitable device which can perform this function. The two bodies, along with the PTO, comprise a wave energy
converter, or WEC.

Signals from an upstream wave detection system 500, responsive to incoming waves 600, are supplied to a wave predictive subsystem 700 whose signals are supplied to a data processor 402 functioning with computer controller 400. Computer/controller 400 may include programming capability and extensive data storage capability 404 all being accessible to the processor 402 for storing data pertaining to the WEC and to assist in the various calculations for generating signals to implement the invention. For example, data pertaining to the height of the waves which actually impinge on the WEC system are stored. Subsystem 700 may be independent of, or be a part of, computer 400. The actual wave sensor may be located at any convenient point enabling the sensing of the incident waves. The wave detection system 500 and the predictive subsystem 700 function to analyze the incoming wave(s) to determine the corresponding anticipated or predicted velocity (V_P) and the force [F_PTO(P)] to which the WEC/PTO should be subjected on the basis of the predicted information and to supply corresponding signals to processor 402.

Signals indicative of the "actual" velocity of the WEC (i.e., the relative movement of the first and second bodies) are also applied to the controller 400. The actual or instantaneous velocity of the WEC (or PTO) refers to the relative velocity of the float and spar. That is, both the float and spar may move with respect to each other, or one may be fixed and the other move. The velocity may refer to any relative motion of the bodies comprising the WEC including linear or rotational motion. Fig. 1 shows a position or velocity sensor 800 coupled to the first and second bodies (100, 200) and having an output coupled to the PTO and the computer/processor 400,402. The sensor 800 may be used to sense the position of the bodies 100, 200 which signals are used (by device 800 or the PTO or the processor) to calculate the actual relative velocity (V_ACT) of the first and second bodies. Alternatively, the actual relative velocity may be obtained directly from components contained within the PTO. V_ACT is used interchangeably to refer to the V_ACT of the WEC or the PTO.

Thus, the computer controller 400 and processor 402 include a data base and data processing capability and are programmed to process incoming ("predicted") and "actual" data to selectively supply control signals to the PTO 300. In accordance with the invention, "actual" or "instantaneous" (as opposed to "predicted") conditions (e.g., velocity) existing within the WEC are sensed (e.g., via an internal or external sensing mechanism) and fed to the control computer 400 which also functions to process information pertaining to the "predicted" wave(s) to generate a force corresponding to these signals to the WEC/PTO.

As discussed above, the "predicted" signals may be in error due to several factors. Thus, though the predicted information is highly desirable and necessary to set up the WEC components to a desired condition for optimizing power capture, Applicant recognized that there will almost always be errors associated with the predicted information, and that the impact of these errors should be minimized. Applicants' invention resides, in part, in the recognition that sensing the actual condition of the WEC/PTO and combining the actual information with the sensed information can be used to produce a more optimal operation, resulting in a greater amount of average power capture and greater efficiency.

The present invention is applicable for use with a broad class of wave energy converters; including WECs having one or more bodies which respond to the waves and/or which may have a rigid connection to the sea-floor and/or which may respond in heave or in another degree of freedom (surge, sway, pitch, roll, yaw); or may respond in any combination of those degrees of freedom; which may include devices such as an oscillating water column or wave-barge. The present invention is also applicable for use with a broad class of power take-off devices, which may include direct linkages or pneumatic or hydraulic linkages.

For ease of explanation, the invention is described, as shown in Fig. 1, with reference to a WEC having: (a) a first body designed to move generally in phase with the waves; (b) a second body designed to move generally out of phase with the waves; where at least one of these two bodies is generally responsive to hydrodynamic wave forces; and (c) a PTO coupled between the two bodies which may be usefully characterized as a device which: (i) generally converts their relative motion into useful energy; and (ii) selectively causes the two bodies to move relative to each other to effectuate an increase in the efficiency of the WEC.

To better understand the invention, a simplified mathematical formulation of a WEC system amenable to numerical modeling is shown in Equation 2 and Equation 3, below. Referring to Figure 1, these equations are obtained assuming that the spar 200 is held fixed and the float 100 tends to move generally in phase with the waves. $$M{\ddot{X}}_{\mathit{WEC}}=F_{\mathit{WAVE}}+F_{\mathit{PTO}}$$ $$F_{\mathit{PTO}}=-B{\dot{X}}_{\mathit{WEC}}$$ Where:

• X_WEC denotes the position of the float relative to the spar,
• F_wave refers to the excitation force acting on the float,
• F_PTO refers to the force exerted by the PTO on the float and the spar, and
• B is some constant of proportionality which specifies a resistive relationship between the force of the PTO and the relative motion of the float and spar.

Equation 2 is a dynamic equation of motion for a single-body wave energy converter. There are two forces on the WEC, the forces of the waves (F_WAVE) and the force (F_PTO) of the power take-off device. A solution of the dynamical equation of motion (Equation 2) requires knowledge of the mathematical formulation of F_PTO (Equation 3.)

It is desirable to rewrite Equation 2 and Equation 3 in the frequency domain. $$V_{\mathit{PTO}}\left(Z_l+Z_{\mathit{PTO}}\right)=F_{\mathit{WAVE}}$$

To derive Equation 4 from Equation 2 and Equation 3, the dynamic variable X_WEC is replaced by the velocity of the PTO (V_PTO), the inertial term MẌ_WEC is replaced with the more general V_PTOZ_I, where Z_I is the intrinsic mechanical impedance, and Equation 3 is rewritten as follows: $${\mathit{F}}_{\mathit{PTO}}={\mathit{Z}}_{\mathit{PTO}}{\mathit{V}}_{\mathit{PTO}}$$

All quantities in Equation 4 may be complex and frequency dependent.
(When complex numbers are used to represent physical quantities, the real parts of the complex numbers are deemed to represent the physical quantities.)

Note that in general, Z_PTO is selected so as to maximize the power transfer between the waves and the PTO. Moreover, for the most general class of Z_PTO values, the calculation of Fpto in Equation 5 requires predictive knowledge of the waves. Once the PTO impedance Z_PTO has been defined, Equation 4 may be solved by known mathematical methods. The velocity V_PTO may be expressed as a function of F_WAVE, Z_I, and Z_PTO as follows: $$V_{\mathit{PTO}}=\frac{F_{\mathit{WAVE}}}{\left(Z_l+Z_{\mathit{PTO}}\right)}$$ With the solution for V_PTO expressed in Equation 6, it is desirable to solve for F_PTO: $$F_{\mathit{PTO}}=Z_{\mathit{PTO}}\frac{F_{\mathit{WAVE}}}{\left(Z_l+Z_{\mathit{PTO}}\right)}$$

Applicant recognized that, based on the predictive information, the PTO force may also be expressed as: $$F_{\mathit{PTO}}\left(P\right)=Z_{\mathit{PTO}}{V}_P$$ Where:

1. (i) F_PTO(P) is the force applied to the PTO by the WEC and/or the force applied by the PTO to the WEC, based on the predicted wave;
2. (ii) V_P is the predicted velocity of the WEC; and
3. (iii) Z _PTO of the PTO is selected to optimize power transfer.

It should be noted that V_P is itself a function of Z_PTO and so Equation 8 is mathematically complicated. However, this representation of F_PTO is useful for clarifying the nature of the present invention.

Applicant further recognized that Equation 8 for F_PTO (P) may be rewritten as: $$F_{\mathit{PTO}}\left(P\right)=Z_{\mathit{PTO}}{V}_P=Z_{\mathit{PTO}}{V}_P-{\mathit{\beta V}}_P+{\mathit{\beta V}}_P=\left(Z_{\mathit{PTO}}-\beta \right)V_P+{\mathit{\beta V}}_P$$

Applicant also recognized that in Equation 9, above, if β satisfies certain constraints (e.g. β is a causal operator) then:

• The (Z_PTO - β)V_P term represents an acausal force (F_A); and
• βV_P represents a causal force (F_C). Where: (a) an acausal force (F_A) as used herein may be defined as a force which requires some predicted information (e.g. wave elevation at some time in the future) to be determined (calculated); and (b) a causal force (F_C) as used herein may be defined as a force which may be calculated without using any predicted information.

$$F_{\mathit{PTO}}\left(N\right)=F_1=\left(Z_{\mathit{PTO}}-\beta \right)V_P+{\mathit{\beta V}}_{\mathit{ACTUAL}}$$

In Equation 10, the causal force, F_C, is determined by calculating βV_ACT; and the acausal force (F_A) as (Z_PTO - β)V_P; where (a) V_ACT is the actual velocity of the WEC; and (b) a value of β can be selected which causes the acausal force (F_A) to be relatively insensitive to wave errors because (Z_PTO - β) is minimized.

Note that F_PTO(N) may also be rewritten as $${\mathrm{F}}_{\mathrm{PTO}}\left(\mathrm{N}\right)=\left({\mathrm{V}}_{\mathrm{P}}\right)\left({\mathrm{Z}}_{\mathrm{PTO}}\right)-\mathrm{\beta }\left({\mathrm{V}}_{\mathrm{P}}-{\mathrm{V}}_{\mathrm{ACT}}\right)$$

Thus, Equation 11 indicates that where there is no wave error, V_P=V_ACT and the term β(V_P-V_ACT) goes to zero and there is no need for correction of the predicted values. Where there is a wave error, implementing the invention compensates for wave error and reduces loss of power capture.

Implementation of these control signals to operate the WEC results in the generation of significantly greater amount of average power when a wave error is present. For a single-frequency case, the average power produced, P_AVG, may be expressed as $P_{\mathit{AVG}}=\frac{1}{2}{V}_{\mathit{ACT}}\overline{F_A}+\frac{1}{2}\mathrm{Re}\left[\beta \right]{\left|V_{\mathit{ACT}}\right|}^2,$ where F_A denotes the complex conjugate of F_A and Re[] denotes the real part of the quantity in brackets.

An implementation of the invention may be explained with reference to Fig. 5. Note that blocks 701-713 in Figure 5 may be part of the processor 402 and programmable controller 400. Sensed wave elevation information is applied to a data base and processor 701 containing information pertaining to the modeling of the WEC and capable of calculating the WEC's reaction to wave elevation changes. The processor 701 can then generate: (a) a PTO force signal, F_PTO(P). and (b) a signal, VP indicative of the anticipated/predicted velocity of the WEC. The impedance (ZPTO) of the PTO is determined for various conditions of the WEC and the waves and data corresponding to Z_PTO is stored in data base section 703 (which may in practice be part of 701) or any other suitable site. The predicted velocity signal (V_P) may then be processed via software/hardware 703 to generate a signal (Z_PTO)V_P. The signal (Z_PTO)V_P can then be modified via software/hardware 705 to generate the product (Z_PTO- β) (V_P). Appropriate values of β may be selected to minimize the term (Z_PTO- β) (V_P). For example, a value of β may be chosen which is close to the value of Z_PTO, at the dominant frequency of the waves, which functions to minimize the signal (Z_PTO- β) (V_P). Note that the minimizing value of β, or any other value of β may be selected for use, depending on predetermined conditions. In a preferred embodiment of the invention, β is a constant over a selected interval, while Z_PTO varies as a function of the wave frequency. Thus, the best value of β is that which cancels out Z_PTO at the dominant frequency of the waves. For a selected value of β the product (V_P)(β) can be obtained by a processing step (see box 707). The product (V_P)(β) can be considered to represent the predicted causal force portion of F_PTO. The next step is to subtract the product (V_P)(β) from F_PTO(P) (see box 709) to obtain the acausal component F_A of F_PTO(P); where F_A = [F_PTO (P)- (V_P)(β)] as the output from the processing step illustrated by box 709. The actual causal component, Fc of the PTO force is obtained by multiplying the actual value of the WEC velocity (V_ACT) by the selected β, as shown by a processing step 711, to produce the product of
[V_ACT(β)]. The actual causal force [Fc =(V_ACT)(β)] may then be added to the predetermined acausal force [F_PTO(P)-(V_P)(β)], as shown in box 713, to produce a modified PTO force which will result in the generation of greater average power when there is wave prediction error. As shown in the figure an output of the processing step represented by box 713 yields: $$\left\{{\mathrm{F}}_{\mathrm{1}}=\left[{\mathrm{F}}_{\mathrm{A}}+{\mathrm{F}}_{\mathrm{C}}\right]=\left[{\mathrm{F}}_{\mathrm{PTO}}\left(\mathrm{P}\right)\cdot \left({\mathrm{V}}_{\mathrm{P}}\right)\left(\mathrm{\beta }\right)\right]+\left({\mathrm{V}}_{\mathrm{ACT}}\right)\left(\mathrm{\beta }\right)=\left[{\mathrm{F}}_{\mathrm{PTO}}\left[\mathrm{P}\right]\cdot \mathrm{\beta }\left({\mathrm{V}}_{\mathrm{P}}-{\mathrm{V}}_{\mathrm{ACT}}\right)\right]\right\}$$

Operating the WEC system in accordance with the invention results in a significant increase in the average power produced by the system when there is a wave error (which is a normal operating condition).

By way of example, various steps to calculate desired values used to practice the invention are shown in Figs. 2, 3 and 4.

Figure 2 is a simplified flow chart illustrating that the sensed incoming waves are processed to determine their properties and any change in their status (see box 21) and that this sampling process occurs at selected interval (see box 29). The range of frequencies present in the wave field is assessed (see box 23). The phase of the waves (not shown) may also be analyzed. Another step (see box 25) in the process is the selection of β which minimizes the function Γ_β(ω) in the freq domain in Equation 26 (see below) over the range of frequencies in the wave field. A further step (see box 27), includes the determination of the function Γ_β(t) in the time domain as per Equation 27 (see below). Another step (see box 29) illustrates that the incoming waves are sampled at selected intervals (e.g., every 5 minutes). The sampling time interval may be increased or decreased depending on the rate of change of the waves. Fig. 2 thus illustrates what may be termed an "outer-loop" control system which may be used to practice the present invention. It highlights the fact that the current invention is best practiced in a manner which depends on the prevailing wave characteristics. That is, there are included means/processes for determining causal and acausal controls which are dependent on the prevailing wave conditions.

Figure 3 is a flow chart illustrating that the amplitude of the sensed incoming waves are processed to determine their properties and any change in their status. In one step (see box 31) the elevation of incoming waves is measured at points upstream from the WEC. Based on a model of wave propagation (see box 33), a prediction can be made of the wave elevation at the WEC corresponding to the sensed incoming waves (see box 35). The information is then processed in the processor (see box 45 in Figure 4)

Figure 4 is a flow chart illustrating basic processing steps for practicing the present invention. The steps include the acquisition of, and determination of, selected characteristics of the predicted wave and computation of causal (Fc) and acausal (F_A) forces, the computation of F_PTO and the application of F_PTO to the PTO. As detailed in Fig. 4, F_A may be obtained by taking the output from the step shown in box 35 in Figure 3 and processing the information as shown in boxes 45 and 46 of Figure 4. This includes computing the acausal force from the convolution integral shown in Equation 28 below to produce F_A. Concurrently, Figure 4 shows that the actual velocity (V_ACT) of the PTO is obtained (see box 47) and applied to the system to compute the causal term, F_c. The information for F_A and Fc are summed (see box 49) to generate the signal corresponding to the force F₁ applied to the PTO in accordance with the invention.

Thus, a method embodying the invention may include:

• (i) Obtaining a mathematical function (termed Γ_β) which is a characteristic of the WEC geometry and the dominant wave conditions;
• (ii) Obtaining wave prediction signals from a wave prediction subsystem;
• (iii) Obtaining an instantaneous measurement of the PTO velocity;
• (iv) Calculating a portion of the desired PTO force as being proportional to a constant (β) multiplied by the PTO velocity; where the value of β is selected as described below with reference to Fig. 9 which shows the real and imaginary parts of the complex PTO impedance for a particular example of a WEC.
• (v) Calculating a second portion of the desired PTO force as being the result of a well-defined mathematical operation depending on the Γβ function from step (i) and the wave prediction from step (ii);
• (vi) Summing together the two PTO forces to obtain the total desired force; and
• (viii) Applying that force to the PTO.

A simplified mathematical model is now discussed which is applicable to both simple and more complicated wave energy conversion systems. This mathematical model of a WEC may be construed to represent the following parts:

1. (a) a wave-responsive (interacting) body which has an intrinsic mechanical impedance, Z_E; (b) a PTO device whose mechanical impedance is selected as Z_PTO, and (c) a response to hydrodynamic wave excitation forces F_WAVE governed by the equation: $$V_{\mathit{PTO}}=\frac{F_{\mathit{WAVE}}}{\left(Z_E+Z_{\mathit{PTO}}\right)}$$ where V_PTO is the velocity of the wave-responsive body, Z_E is the intrinsic mechanical impedance, and F_WAVE is the hydrodynamic wave excitation force acting on the wave-responsive body. The hydrodynamic wave excitation force F_WAVE may be also expressed as $${\mathit{F}}_{\mathit{WAVE}}\equiv {\mathit{\eta \chi }}_{\mathit{E}}$$ Where:
   • η is the instantaneous surface elevation of the water at the location of the WEC which would be measured in the absence of the WEC, and
   • χ_E is the wave excitation force response. The wave excitation force response is a function of the water depth and the body geometry. Note that in Equation 13, V_PTO, may represent a rotational degree of freedom, so that F_WAVE would represent a hydrodynamic torque, and V_PTO would represent an angular velocity.

The force F_PTO in the PTO is equal to $$F_{\mathit{PTO}}=Z_{\mathit{PTO}}{V}_{\mathit{PTO}}$$

The instantaneous power absorbed by the PTO can be expressed as $$P_{\mathit{AVG}}=V_{\mathit{PTO}}\overline{F_{\mathit{PTO}}}$$ Where F_PTO is the complex conjugate of F_PTO.

For a wave absorber of the type shown in Figure 1, Equation 12, Equation 13, Equation 14, and Equation 15 may be derived for the case where the spar 200 is held fixed, or the case where the spar 200 is allowed to move. For the case where the float 100 moves in phase with the waves and spar 200 is held fixed, the mechanical impedance Z_E can be expressed as $$Z_E=\mathit{i\omega M}\left(\omega \right)+B\left(\omega \right)-\frac{i}{\omega }C$$

Where: ω is the angular frequency, M is the mass of the body 100 including displaced mass and hydrodynamic added mass,, B is the hydrodynamic radiation damping of body 100, and C is the hydrostatic restoring force, defined as (g)(S)(ρ), where ρ is the density of water, g is the acceleration of gravity, and S is the waterplane area of body 100.

The quantities M, B, and C are functions of the shape of the body, and the water depth, and may be determined by appropriate modelling.

For the case where the spar 200 is not held fixed, Equation 12, Equation 13, Equation 14, and Equation 15 are still valid. The intrinsic impedance of the mechanical system is no longer specified by Equation 16 and may be determined using known techniques.

As already discussed, for Z_PTO to provide optimal power conversion efficiency, the PTO impedance Z_PTO may be selected to be equal to the complex conjugate of the intrinsic impedance Z_E, such that: $${\mathit{Z}}_{\mathit{PTO}}={\bar{\mathit{Z}}}_{\mathit{E}}$$ Given this expression for Z_PTO, the optimal PTO velocity may be expressed as $$V_{\mathit{PTO}}=\frac{F_E}{\left(Z_E+Z_{\mathit{PTO}}\right)}=\frac{F_E}{2\mathrm{Re}\left[Z_E\right]}$$ Where Re[] denotes the real part of the quantity in brackets. Noting that F_E = ηχ_E and defining $$\mathrm{\Lambda }\equiv \frac{{\chi }_E}{2\mathrm{Re}\left[Z_E\right]}$$ The following expression may be obtained for the optimal PTO velocity: $$V_{\mathit{PTO}}^{\mathit{OPTIMAL}}=\mathrm{\Lambda }\mathit{\eta }$$

Note that Equation 20 defines an expression which is not causal as the solution of ${\mathit{V}}_{\mathit{PTO}}^{\mathit{OPTIMAL}}$ requires foreknowledge of the wave elevation. Mathematically, this is because this equation must be represented as a convolution, and since the function A has non-zero values for negative values of time, calculation of V_PTO requires knowledge of the wave elevation ^η for positive values of time (i.e. in the future.)

If the wave elevation is known sufficiently far into the future so that ${\mathit{V}}_{\mathit{PTO}}^{\mathit{OPTIMAL}}$ may be calculated, then the PTO may be controlled so that V_PTO is forced to be ${\mathit{V}}_{\mathit{PTO}}^{\mathit{OPTIMAL}}\mathrm{.}$ This type of control, expressed in Equation 20 may be termed a feedback law, since a feedback loop is used to control the PTO velocity V_E.
It is also possible to construct a "feedforward" law, by defining Γ₀ =ΛZ_PTO, to produce: $${\mathrm{\Gamma }}_0\equiv \frac{{\chi }_E}{2\mathrm{Re}\left[Z_E\right])}\overline{Z_E}\mathrm{.}$$ This can be used to generate the following expression for the optimal PTO force: $$F_{\mathit{PTO}}^{\mathit{OPTIMAL}}={\mathrm{\Gamma }}_0\eta$$ Equation 22 defines a control law for the WEC which is acausal like Equation 20. This method may be called a feedforward law because the output of the control algorithm calculation is the force to be applied to the PTO with no closed-loop control. This method is also acausal because calculation of $F_{\mathit{PTO}}^{\mathit{OPTIMAL}}$ in Equation 22 requires knowledge of the wave elevation at future times.
Equation 20 and Equation 22 are both written as frequency domain multiplications. To put these equations into the time domain, the underlying operators (A, in the case of Equation 20 and Γ₀ in the case of Equation 22) must be transformed from the frequency domain to the time domain by way of a Fourier transform, and then the multiplication must be carried out as a convolution integral.
An explicit representation of Γ₀ in the frequency domain is shown below in Equation 23. The Fourier transform operation, which results in an expression for Γ₀ in the time domain is shown explicitly in Equation 24 below. With the explicit expression for Γ₀ in the time domain, and a time-series dataset of wave height η(t), it is possible to carry out the convolution integral and thus calculate the force (F_PTO) to be applied to the PTO. This convolution integral is shown explicitly in Equation 25. It should be noted that the limits of the integral in Equation 25 should extend over all times. However, as indicated in Figure 8, the Γ₀ function decreases to relatively small values, and the integral may be truncated accordingly at times t_s - T_Γ+ and t_s + T_Γ-. $${\mathrm{\Gamma }}_0\left(\omega \right)=\frac{{\chi }_E}{\left(Z_E+Z_{\mathit{PTO}}^{\mathit{optimal}}\right)}{Z}_{\mathit{PTO}}^{\mathit{optimal}}$$ $${\mathrm{\Gamma }}_0\left(t\right)=\int {\mathrm{\Gamma }}_0\left(\omega \right)e^{-\mathit{iax}}\mathit{d\omega }$$ $$F_0\left(t_s\right)={\displaystyle {\int }_{t_s-T_{r+}}^{t_s+T_{r-}}}n\left(\tau \right){\mathrm{\Gamma }}_0\left(t_s-\tau \right)\mathit{d\tau }$$

An important aspect of the invention is applicant's recognition that the prior art optimal feedforward force $F_0=F_{\mathit{PTO}}^{\mathit{OPTIMAL}}$ (defined in Equation 22) may be replaced by a new optimal feedforward force F₁ which may be expressed as the sum of two forces (F_A and F_C), which together sum to the optimal feedforward force, and which individually have desirable characteristics. The force F_C which can be expressed as βV_ACT and is defined in Equation 29 below has the desirable characteristic of being causal (and hence insensitive to wave prediction error). The acausal force, F_A, which can be expressed as F_PTO(P)- βV_P and is defined in Equation 28 below has the desirable characteristic of being less sensitive to wave prediction error than F₀ where F₀ is the optimal PTO force in accordance with the prior art which is calculated solely from predicted wave information. Under idealized (zero wave error) conditions, F₀ and F₁ are equivalent. Under the anticipated condition that there is wave prediction error, F₁ has the desirable characteristic that it is less sensitive to wave prediction error than F₀. As a result, a new function, called Γ_β is introduced as a replacement for the Γ₀ function, and having the following definition: $${\mathrm{\Gamma }}_{\beta }\left(\omega \right)=\frac{{\chi }_E}{\left(Z_i+Z_{\mathit{PTO}}^{\mathit{optimal}}\right)}\left(Z_{\mathit{PTO}}^{\mathit{optimal}}-\beta \right)\mathrm{.}$$

In practice, the quantity β is part of the causal relationship, and the resulting expression for Γ_β(ω) is made as small as possible in the frequency range of interest. Figure 9 shows an example of the mechanical impedance Z_E for a particular WEC system. The optimal PTO impedance, $Z_{\mathit{PTO}}^{\mathit{optimal}},$ is the complex conjugate of Z_E. With this knowledge, β may be chosen so as to make Γ_β(ω) as small as possible in the frequency range of interest. It may be deduced that appropriate selection of β will assure that Γ_β is smaller than Γ₀ in the frequency range of interest. It may be deduced that different values of β may be desirable depending on the frequencies of the waves of interest. Once Γ_β(ω) is defined in the frequency domain, it is desirable to carry out a Fourier transform to express Γ_β(t) in the time domain. This Fourier transform operation is defined in Equation 27 (in analogy with Equation 24). The convolution integral which results in the calculation of F_A is shown explicitly in Equation 28. $${\mathrm{\Gamma }}_{\beta }\left(t\right)=\int {\mathrm{\Gamma }}_{\beta }\left(\omega \right)e^{-\mathit{i\omega l}}\mathit{d\omega }\mathrm{.}$$ $$F_A={\displaystyle {\int }_{t_s-T_{r+}}^{t_s+T_{r-}}}\eta \left(\tau \right){\mathrm{\Gamma }}_{\beta }\left(t_s-\tau \right)\mathit{d\tau }\mathrm{.}$$ Fc may be calculated as follows $$F_C={\mathit{\beta V}}_{\mathit{Act}}\left(t_s\right)\mathrm{.}$$ Summing F_A and Fc results in F_PTO, or F₁, in accordance with the invention. $$F_1=F_A+F_C\mathrm{.}$$

A prescriptive outline of the feedforward control law according to the present invention is given below:

1. 1) Tasks to perform once per wave condition
   1. a. Calculate Z_E, Intrinsic mechanical impedance of WEC
   2. b. Calculate χ_E, Wave excitation force response
   3. c. Calculate $Z_{\mathit{PTO}}^{\mathit{optimal}},$ Optimal mechanical impedance of PTO
   4. d. Select a value of β which will minimize Γ_β(ω) (as per Equation 26) over the frequency range of interest (e.g., over the frequency range of the waves) subject to the constraint that β be constant or β be a causal operator
   5. e. Calculate Γ_β(ω) according to Equation 26
   6. f. Calculate Γ_β(t) in the time domain according to Equation 27
   7. g. Determine T_Γ-, T_Γ+: Minimum, maximum support of Γ_β(t) (See Figs. 8,9,10)
2. 2) Tasks to perform at every time step t_s
   1. a. Determine prediction of wave elevation T_Γ- seconds into the future. [Note that now we have an expression for the wave elevation η as a function of time from t_s - T_Γ+ to t_s + T_Γ-]
   2. b. Calculate acausal force term from convolution integral (Equation 28)
   3. c. Determine instantaneous actual PTO velocity V_Act
   4. d. Calculate causal force term F_C = βV_Act(t_s)
   5. e. Calculate F₁ = _FA + F_C
   6. f. F₁ is the optimal PTO force in accordance with the present invention. Control the PTO to apply F₁ to the PTO.

Figure 6 shows a simplified schematic of a WEC with velocity sensing means. The wave-responsive bodies 100 and 200 move in response to waves, leading to motion of the Power Take Off system (PTO) 300. The PTO is comprised of a non-rectified linear-to-rotary conversion mechanism, consisting of a hydraulic piston 301, hydraulic tubing 302, and a hydraulic motor 303. The rotary motor 303 is colocated on a shaft 304 with a mechanical-to-electrical converter (generator) 305. A load circuit 306 includes a resistor (load) 308,connected across the generator 305 and a voltage sensing device 309 connected across the load. There is a linear relationship between the relative velocity between one body, 100, (e.g., one of a float and spar) and the other body, 200, (e.g., the other one of a float and spar) and the voltage produced by generator 305 and sensed by voltage sensor 309. The relationship between measured voltage and relative velocity between float and spar may be determined as a function of the characteristics of the components of the PTO. A cable 311 transmits the voltage signal to the control computer 400, thus allowing the control computer to determine the actual velocity of the PTO. Thus, Fig. 6 illustrates a means for measuring the relative velocity between float and spar.

Figure 7 shows a simplified schematic of a WEC with position sensing means.. The wave-responsive bodies 100 and 200 move in response to waves. The components 550 and 551 comprise a position sensing mechanism. For a known position sensing mechanism, 551 represents a wire under tension between body 100 and body 200. Component 550 contains a tensioned spool which can take in (or let out) wire 551 as the relative position between body 100 and body 200 changes. Component 550 contains means for detecting length of wire 551 paid out, and transmits this information via a cable 311 to the control computer 400, thus allowing the control computer to determine the actual distance between the body 100 and the body 200. The control computer 400 may differentiate this signal to determine the actual velocity of the PTO. Figs: 6 and 7 are meant to illustrate that any suitable actual velocity sensing means may be used to practice the invention.

Figure 8 shows schematic representations of the force response transfer functions Γ₀ (t) and Γ_β(t). These functions Γ₀(t) and Γ_β(t) tend to be relatively large in magnitude near the t=0 axis, and tend towards zero in magnitude as t approaches + or - infinity. Consequently, there are positive and negative values of t beyond which the magnitude of functions Γ₀(t) and Γ₀(t) is sufficiently close to zero that in practice, it is not necessary to carry out the convolution integral (Equation 28) beyond these values of t. These limits are indicated as T_Γ- for negative values of t and as T_Γ+ for positive values of t.

Figure 9 shows an example of the optimal PTO impedance (Z_PTO) for a wave energy converter. Real and imaginary parts are shown in the top and bottom plots, respectively. It can be seen that the imaginary part passes through zero in the vicinity of 0.11 Hz for a given WEC design.

Figure 10 shows a frequency domain graph of the prior art functions Γ₀(ω) and the corresponding function, Γ_β(ω), according to the invention, for a real and constant value of β. The function Γ_β(ω) is defined in Equation 26 as:

• ${\mathrm{\Gamma }}_{\beta }\left(\omega \right)=\frac{{\chi }_E}{\left(Z_i+\overline{{\mathit{Z}}_{\mathit{PTO}}^{\mathit{optimal}}}\right)}\left({\mathit{Z}}_{\mathit{PTO}}^{\mathit{optimal}}-\beta )\right)\mathrm{.}$ The function Γ_β(ω) is based on
  1. (a) the characteristics of a particular wave energy converter such that the graph of ${\mathit{Z}}_{\mathit{PTO}}^{\mathit{optimal}}$ for that WEC is shown in Figure 9; and
  2. (b) a value of β of approximately 500 kNs/m.

It may be seen from the graph of ${\mathit{Z}}_{\mathit{PTO}}^{\mathit{optimal}}$ (Figure 9) that, for β = 500 kNs/m, the value of $\left({\mathit{Z}}_{\mathit{PTO}}^{\mathit{optimal}}-\beta \right)$ is zero at approximately 0.2 Hz. Likewise, the function Γ_β(ω) is also minimized at 0.2 Hz. This value of β, selected in conjunction with the practice of the invention, thus minimizes sensitivity of the WEC system to wave error for waves at the frequency of approximately 0.2 Hz. Other values of β may be selected, in accordance with the invention, so as to minimize sensitivity of the WEC system at other dominant wave frequencies.

Figures 11, 12, 13, and 14 are waveforms diagram used to illustrate the invention and to assist in its explanation.

Figure 11 shows optimal control intermediate results using the prior art. This figure applies to a condition of zero wave error which is defined as the condition where the anticipated or predicted wave is identical to the wave that impinges on the WEC:

• a--Waveform A in Fig. 11, depicts the elevation (E) of a wave impinging on the location of the WEC.
• b-- Waveform B in Fig. 11 depicts the velocity of the WEC (the moving part or parts of the WEC) in response to Waveform A.
• c- Waveform C in Fig. 11 depicts the PTO force, F_PTO. of the WEC's PTO. In effect, the wave elevation and the F_PTO define the velocity of the moving body (WEC).
• d- Waveform D in Fig. 11 depicts the power generated (P) by the WEC;
  P is equal to velocity times (F_PTO); where the velocity is (V_ACT).

Figure 12 shows optimal control intermediate results using the present invention. This figure also applies to a condition of zero wave error. Waveforms A and B in Figure 12 depict, respectively, the elevation (E) of a wave impinging on the location of the WEC and the corresponding velocity of the WEC. Waveform C in Figure 12 depicts the PTO force, F_PTO, of the WEC's PTO and that F_PTO may be expressed in terms of a causal force Fc and an acausal force F_A;
where: $${\mathrm{F}}_{\mathrm{PTO}}=\left({\mathrm{V}}_{\mathrm{P}}\right)\left({\mathrm{Z}}_{\mathrm{PTO}}-\mathrm{\beta }\right)+\left({\mathrm{V}}_{\mathrm{P}}\right)\left(\mathrm{\beta }\right)$$ $${\mathrm{F}}_{\mathrm{A}}=\left({\mathrm{V}}_{\mathrm{P}}\right)\left({\mathrm{Z}}_{\mathrm{PTO}}-\mathrm{\beta }\right)$$ and $$\mathrm{Fc}=\left({\mathrm{V}}_{\mathrm{P}}\right)\left(\mathrm{\beta }\right)$$ It should be noted that the causal and acausal forces (in subplot 3 - waveform C) add up to the optimal force in subplot 3.
Waveform D in Figure 12 depicts the power generated (P) by the WEC;
P is equal to V_P multiplied by (Fc + F_A); which is equal to the power of the system of Fig. 11. It should be noted that the causal and acausal power contributions add up to the optimal power profile in subplot 4, waveform D.

Figure 13 shows control using the prior art in the presence of wave error. Thus this figure illustrates the condition where the actual wave impinging on the WEC is different than the predicted wave, as shown in waveform A. For this condition it is demonstrable that the F_PTO and the actual velocity will be different than for the conditions of Fig. 12. Thus, the predicted velocity V_P will differ from the actual velocity V_ACT. However, as per the prior art, continuing to maintain F_PTO = (V_P )(Z_PTO); i.e., not differentiating between causal and acausal forces and maintaining F_PTO = (V_P)(Z _PTO); i.e., not differentiating between causal and acausal forces and maintaining Fc = (V_P)( β) and F_A = (V_P)(Z_PTO - β) results in the generation of much less power by the WEC when there is a wave error (i.e., the actual wave is different than the predicted wave). For example, 412kW of power are generated, as shown in waveform D of Fig. 13, versus 563 kW as shown in waveform D of Figs. 11 and 12. It should be noted that the causal part of the force is proportional to the predicted velocity.

Figure 14 shows control using the present invention in the presence of wave error. Thus, this figure illustrates the condition where the actual wave impinging on the WEC is different than the predicted wave, as shown in waveform A. But, in accordance with the invention, for the conditions shown in Fig. 14, F_PTO is processed as having a causal (Fc) force and an acausal force (F_A); where Fc is set equal to β multiplied by V_ACT and F_A = (V_P)(Z _PTO - β). The power generated by the WEC in accordance with this processing arrangement is significantly higher than for the uncorrected arrangement of the prior art. This may be demonstrated by comparing the average power (511 kW) produced as shown in waveform D of Fig. 14 versus that shown in waveform D of Fig. 13. It should be noted that the causal part of the force is based on the actual/observed WEC velocity, not the predicted velocity.

Figure 15 shows a summary of results for converted power comparing the invented with the "known" method for a particular WEC configuration under a particular wave condition. The x-axis of the graph represents the degree of wave error, and the y axis represents the power produced by a given wave energy conversion system in the presence of a 7 second wave. The y-axis is normalized relative to the power produced by "proportional control," , which is a control law that in no way relies on predictive wave information Thus, values less than "1" indicate failure for a control law: if a control law does not produce more power than "proportional control" control, then it is providing no benefit. The three curves in the graph represent the power produced in each of three different control schemes. Note that all three control schemes are in agreement where there is no wave error (e.g. on the left side of the graph, where the x-axis is zero.) The control scheme denoted by the "A" curve represents a "known" control scheme. The Γ₁₀₀₀ and Γ₂₀₀ ,curves represent power results for two of the possible control schemes. Note that the Γ₁₀₀₀ control law is providing benefits relative to "proportional control," until there is an error of 10% in the WEC position. The A control law provides benefits relative to "proportional control," only for wave error of less than 5%. Thus, there is an improvement of over 100% for the Γ₁₀₀₀ relative to the result for the "A" control law. Symbol Definition M Mass of moving body in simple model of WEC X_WEC Position of moving body in simple model of WEC V_PTO Velocity of power take-off (PTO) in model of WEC F_WAVE Wave-induced force on moving body in model of WEC F_PTO PTO force acting on moving body in model of WEC Z_E Intrinsic mechanical impedance of a wave-interacting body which is part of idealized wave energy converter. Z_PTO Mechanical impedance of a power take-off which is part of idealized wave energy converter. ${\mathit{Z}}_{\mathit{PTO}}^{\mathit{optimal}}\mathrm{.}$ Optimal value of mechanical PTO impedance F_E Hydrodynamic excitation forces acting on the wave-interacting body. η Instantaneous surface elevation of the water at the location of the WEC which would be measured in the absence of the WEC χ_E The wave excitation force response Λ Operator relating incident wave elevation to $V_{\mathit{E}}^{\mathit{OPTIMAL}}$ $V_{\mathit{E}}^{\mathit{OPTIMAL}}$ Velocity of equivalent system assuming optimal control Γ₀, Γ₀(ω) Operator relating incident wave elevation to optimal force according to prior art $F_{\mathit{E}}^{\mathit{OPTIMAL}}$ Force of PTO assuming optimal control Z_E Complex conjugate of Z_E T_Γ- Minimum support of function Γ(t) (e.g. minimum value of t for which Γ(t) has a value substantially different from 0) T_Γ+ Maximum support of function Γ(t) (e.g. maximum value of t for which Γ(t) has a value substantially different from 0) t_s Time at which a control algorithm step is carried out. F₀, F₀(t_s) Force to be applied to the PTO at time t_s to provide optimal control according to the prior art. Γ_β, Γ_β(ω) Operator relating incident wave elevation to acausal component of optimal force according to invention β Constant causal resistance relating V_E to causal component of optimal force according to invention Γ_β(t) Time domain representation of operator relating incident wave elevation to acausal component optimal force according to invention Γ₀(t) Operator relating incident wave elevation to optimal force according to prior art, time domain representation F₁, F₁(t_s) Force to be applied to the PTO at time t_s to provide optimal control according to the invention.